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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

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BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

In Problems 4-6, use the function y = 3 x 5 5 x 3 + 2 .

Over what intervals is the graph of this function concave up?

To determine

To calculate: The intervals for which the graph of the function y=3x55x3+2 is concave up.

Explanation

Given Information:

The provided equation is y=3x55x3+2.

Formula used:

Concavity of f(x) can be found from the sign of f(x). Because f(x) is the first derivative of f(x), wherever the graph of f(x) is increasing, it follows that f(x)>0.

Thus f(x)>0 and f(x) is concave up.

Calculation:

Consider the provided equation,

y=3x55x3+2

Calculate the first derivative of the above equation with respect to x:

y=15x415x2=15x2(x21)=15x2(x1)(x+1)

Differentiate with respect to x again:

y=60x330x=30x(2x21)

Factorize (2x21) and simplify.

30x(2x21)=30x×2(x212)=60x(x2(12)2)=60x(x12)(x+12)

Now, concavity of f(x) can be found from the sign of f(x). Because f(x) is the first derivative of f(x), wherever the graph of f(x) is increasing, it follows that f(x)>0

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